After throwing interceptions on consecutive drives to allow the Falcons to surge ahead 14-10 in the fourth, quarterback Brees capped a 90-yard march downfield with a six-yard touchdown pass to Jimmy Graham with 3:28 remaining.
The Saints stopped Atlanta on its next possession and then ran down the clock to hand the Falcons their first home loss of the season.
"I was telling myself 'you have to put that behind you,' " Brees told reporters of his two late miscues.
"You had to in a game like this... stay composed and find a way to convert and go down and score. In the end, we found a way to win." Brees finished with 302 yards passing and running back Pierre Thomas added 63 yards rushing and one score.
Defending Super Bowl champion New Orleans (11-4) moved within one game of the Falcons in the NFC South but Atlanta will secure their first division title since 2004 with either a win or a Saints loss in the Week 17 season finale.
Atlanta (12-3) would have clinched home field advantage throughout the NFC playoffs with a victory but instead had its eight-game winning streak snapped.
On their final possession, Atlanta drove the ball to its own 43 and elected to punt with just under three minutes left and the team never got the ball back.
"I thought it was the right thing to do," Falcons coach Mike Smith said. "I felt like the way we had been playing defensively - and we had the two-minute warning. Unfortunately we didn't get it done at the end." The Falcons fell behind 10-0 in the second quarter but got a TD pass from Matt Ryan to Roddy White to draw closer before halftime.
Atlanta took the lead when defensive lineman Chauncey Davis intercepted a Brees shovel pass and ran 26 yards for the score.
Fellow lineman John Abraham also intercepted Brees on New Orleans' next possession but the Falcons could not convert it into points.
QB Ryan threw for only 148 yards as Atlanta amassed just 215 total yards.
